Much of the data collected during the monitoring of cyber and other infrastructures is structural in nature, consisting of various types of entities and relationships between them. The detection of threatening anomalies in such data is crucial to protecting these infrastructures. We present an approach to detecting anomalies in a graph-based representation of such data that explicitly represents these entities and relationships. The approach consists of first finding normative patterns in the data using graph-based data mining and then searching for small, unexpected deviations to these normative patterns, assuming illicit behavior tries to mimic legitimate, normative behavior. The approach is evaluated using several synthetic and real-world datasets. Results show that the approach has high truepositive rates, low false-positive rates, and is capable of detecting complex structural anomalies in real-world domains including email communications, cellphone calls and network traffic.